I bumped into a claim I am not understanding completely about Euler products. Is it true that \begin{align} &\prod_p (1 + a(p) p^{-s} + a(p^2) p^{-2s} + \cdots) \\ &\hspace{2cm}= \prod_p (1 + b(p) p^{-s} + b(p^2) p^{-2s} + \cdots)\\ &\hspace{2cm}\qquad \times \prod_p (1 + (a(p^2)-b(p^2)) p^{-2s} + (a(p^3)-b(p^3)) p^{-3s} + \cdots)? \end{align}
And also, a second question that may indicate I am missing something: if $f(p) = a(p) + b(p)$ (just for $p$, not for all $p^m$, and even with this I do not see why it should be true), do we have \begin{align} &\prod_p (1 + f(p) p^{-s} + f(p^2) p^{-2s} + \cdots) \\ &\hspace{1cm}= \prod_p (1 + a(p) p^{-s} + a(p^2) p^{-2s} + \cdots) \prod_p (1 + b(p) p^{-s} + b(p^2) p^{-2s} + \cdots)? \end{align} i.e. the Dirichlet series associated to $f$ factorizing into the product of the Dirichlet series of $a$ and $b$? I thought it was rather the case when $f = a \star b$, the arithmetic convolution of both. Is it better if we assume all these functions to be completely multiplicative?
I do not see the magic making it work, so I ask for help. Even for the $p^2$ term it does not seem to be true?